The book under review is concerned with a conjecture of G. Lusztig on the based ring of a two-sided cell \(\Omega\) in an affine Weyl group \(W\). The conjecture proposes the existence of an isomorphism between the based ring of \(\Omega\) and a certain equivariant \(K\)-group [see J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 2, 297-328 (1989; Zbl 0688.20020)]. Since its importance in the representation theory, it is desirable to provide a verification for this conjecture. Thus a neat and correct proof for the conjecture is highly recommended, even only for some special cases. The author verified the conjecture in various special cases (i.e., all the rank 2 cases, the lowest two-sided cell case and the second highest two-sided cell case) in his previous book [Representations of affine Hecke algebras, Lect. Notes Math. 1587 (1994; Zbl 0817.20051)].
The aim of the present book is to verify the conjecture in the case where \(W\) is of type \(\widetilde A_{n-1}\), \(n>1\). Based on the knowledge of the permutational description for the cells of the group \(W\) [see Jian-yi Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lect. Notes Math. 1179 (1986; Zbl 0582.20030)], the author first uses two-sided star operations to reduce the problem to the case where one needs only consider the set \(\Gamma_\lambda\cap\Gamma^{-1}_\lambda\) with \(\Gamma_\lambda\) a certain special left cell of \(W\) corresponding to a partition \(\lambda\). Then he establishes a bijection between the sets \(\Gamma_\lambda\cap\Gamma_\lambda^{-1}\) and \(\text{Dom}(F_\lambda)\) by defining a map \(\varepsilon\), \(\text{Dom}(F_\lambda)\) a certain parameter set of the isomorphism classes of irreducible rational representations of a reductive algebraic group \(F_\lambda\). Finally, the author proves that the based ring of \(\Gamma_\lambda\cap\Gamma_\lambda^{-1}\) is isomorphic to the representation ring of \(F_\lambda\) by using the map \(\varepsilon\).